This was stated in my thermo dynamics lecture today and I tried to ask my lecturer why it was not just defined as 0 since 0.01 seems weirdly specific. She was mentioning something about the order in which the celcius and kelvin scales came about but I am still uncertain about the answer.
At very low pressures (less than 1/1000 of an atmosphere) water goes directly from a solid to a gas phase as temperature increases. At about 0.006 atmospheres, the pressure of the triple point, the pressure allows for a liquid phase to exist at intermediate temperatures between solid and gas.
The temperature at which the transition from solid to liquid happens falls very gradually as pressure increases above that of the triple point. As a result, at one atmosphere of pressure ice will melt at a slightly lower temperature than that of the triple point. The difference is about 0.01 C. So, in defining 0 C, one can pick either the temperature of the triple point, or the temperature at which ice melts at one atmosphere, but you can't pick both.
Now, the triple point is more easily controlled in a laboratory, so it is the safer place to set an exact standard. But broader considerations are that you want the standard pressure, one atmosphere, melting point to be close to zero - because that is what people are used to. So the exact standard is set at the triple point, but it involves a little fudge to make the one atmosphere point be closer to zero.